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Let χ be the primitive Dirichlet character of conductor 49 defined by χ(3)=ζ for ζ a primitive 42nd root of unity. We explicitly compute the slopes of the U7 operator acting on the space of overconvergent modular forms on X1(49) with weight k and character χ7k−6 or χ8−7k, depending on the embedding of ℚ(ζ) into ℂ7. By applying results of Coleman and of Cohen and Oesterlé, we are then able to deduce the slopes of U7 acting on all classical Hecke newforms of the same weight and character.

We prove that near the boundary of weight space, the 2-adic eigencurve of tame level 1 can be written as an infinite disjoint union of “evenly-spaced” annuli, and on each annulus the slopes of the corresponding overconvergent eigenforms tend to zero.